Using the compactness of the class of \(\mathsf {RCD}(K,N)\)  spaces with uniformly bounded diameter under measured Gromov-Hausdorff convergence [1]}, [2]}, the stability properties of \(\lambda _{1}\)  and \(h(X)\)  under such convergence [2]}, [4]}, and the fact that no \(\mathsf {RCD}(K,N)\)  space can split isomorphically a Gaussian (since the former is measure-doubling while the latter is not), we obtain the next dimensional improvement of (REF ) by a straightforward argument by contradiction (notice that \(K>0\)  implies a uniform upper bound on the diameter thanks to the Bonnet-Myers theorem [1]}).
